Definition of spatial rectangular coordinate system;
After passing through the fixed point o, make three mutually perpendicular number axes, all of which take o as the origin and generally have the same length unit. These three axes are called X axis (horizontal axis), Y axis (vertical axis) and Z axis (vertical axis) respectively. Generally called coordinate axis, X axis and Y axis are usually arranged on a horizontal plane, while Z axis is a vertical line; Their positive direction should conform to the right-handed rule, that is, the right hand holds the Z axis. When the four fingers of the right hand turn from the positive X axis to the positive Y axis at π/2 angle, the thumb points to the positive direction of the Z axis, so the three coordinate axes form a spatial rectangular coordinate system, and the point O is called the coordinate origin.
1, right-hand rectangular coordinate system
(1) Rules for establishing right-hand rectangular coordinate system: X-axis, Y-axis and Z-axis are perpendicular to each other and point to thumb, forefinger and middle finger of the right hand respectively;
(2) The method and steps of taking the coordinates P(x, y, z) of a known point as a point (path method);
In the positive direction of the x axis (x >；; 0) or negative direction (x0) or negative direction (y0) or negative direction (z
(3) Location coordinate method of known points:
If the three planes passing through P are perpendicular to the X-axis, Y-axis and Z-axis respectively, then the coordinates of points A, B and C on the X-axis, Y-axis and Z-axis respectively are A, B and C, then (A, B and C) are the coordinates of point P..
2. The points on the X axis can be expressed as (a, 0, 0), (0, b, 0) and (0, 0, c) respectively.
Points in the coordinate planes xOy, xOz and yOz can be expressed as (a, b, 0), (a, 0, c) and (0, b, c) respectively.
3. The coordinate of the point P(a, b, c) relative to the X axis is (a, -b,-c);
The coordinates of point P(a, b, c) with respect to the Y axis symmetry point are (-a, b,-c);
The coordinates of point P(a, b, c) with respect to the Z -axis symmetry point are (-a, -b, c);
The symmetry point of point P(a, b, c) with respect to coordinate plane xOy is (a, b,-c);
The symmetry point of point P(a, b, c) with respect to coordinate plane xOz is (a, -b, c);
The symmetry point of the point P(a, b, c) with respect to the coordinate plane yOz is (-a, b, c);
Symmetrical points (-a, -b, -c) of point P(a, b, c) relative to the origin.
4. Given two points in the space P(x 1, y 1, z 1) and Q(x2, y2, z2), the midpoint coordinates of the line segment PQ are
5. Distance formula between two points in space
Given two points P(x 1, y 1, z 1) and Q(x2, y2, z2) in the space, the distance between these two points is a special point A(x, y, z) and the distance from the origin O is
6. The spherical equation with C(x0, y0, z0) as the center and r as the radius is
Especially, the spherical equation with the origin as the center and R as the radius is x2+y2+z2=r2.
Exercise questions:
Multiple choice question:
1. In the space rectangular coordinate system, the point P(x, y, z) is known, and the following four statements are given: ① The coordinate of point P about the X-axis symmetry point is (x, -y, z); ② The coordinate of point P about the yOz plane symmetry point is (x, -y,-z); ③ The coordinate of point P about the Y-axis symmetry point is (?
A.3B.2C. 1D.0
2. If A( 1, 1, 1) and B (-3, -3, -3) are known, the length of the line segment AB is ().
A: 43
b23
C.42
Cao32
3. If A( 1, 2,3), b (3,3, m), C(0,-1, 0) and D(2,-1,-1) are known, then ().
A.|AB| >|CD|
B.|AB|